On Quadratic Periodic Points of Quadratic Polynomials
نویسندگان
چکیده
We focus on a very specific case of the Uniform Boundedness Conjecture, namely, bounding the number of possible c such that the quadratic polynomial φc(z) = z2 + c has quadratic periodic points of some small period. We show that there are infinitely many rational c with quadratic 4-cycles, with all such c completely understood; and only finitely many rational c with quadratic 5-cycles (we conjecture that there are none). We also present a general theorem, together with a more powerful conjecture, that may be useful for reducing the search of quadratic cycles (or higher) to that of rational cycles. Computational data have been produced to support our conjectures.
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